Download 2.1 Density Curves and the Normal Distributions

2.1 Density Curves and the
Normal Distributions
Grab warm-up and start!
Strategies in Chapter 1
New Strategy:
Sometimes the overall pattern of a large number of
observations is so regular that we can describe it by a
smooth curve.
Density Curves
Characteristics:
Always on or above horizontal axis
Has an area of 1 underneath the curve
What are they:
o
o
o
o
Density curves are mathematical models for distributions.
Outliers are not described by the curve.
All curves are approximations.
They are idealized.
EXAMPLE
Familiar Shapes of Models
or Density Curves
Right Skewed
Left Skewed
Symmetric/Bell-shaped
Mean and Median of Density Curves
The median of a density curve is the equal areas point.
The mean is at the balance point (the point where the
curve would balance if it were made of solid material).
The mean and median of symmetric curves are at
the center.
Normal Distributions
 Normal distributions can be described with a
density curve that is symmetric, single peaked,
and bell-shaped.
 Changing the mean without changing standard
deviation moves the mean along x-axis without
changing the spread.
 Standard deviation controls the spread of the
curve. (how flat or peaked it is!)
What is a normal distribution?
 Normal distributions are a family of
distributions that have the same general
shape.
 They are symmetric with scores more
concentrated in the middle than in the
tails.
 bell shaped
 two parameters: the mean () and the
standard deviation ().
 Inflection points
 The points at which the graph changes
concavity (curvature)
 Inflection points are  units on either side
of the mean .
Notation
Idealized Distribution
Notation:
VS.
Mean: µ (greek letter mu)
Standard deviation:
σ (greek letter sigma)
Normal Distributions: N(µ, σ)
Sample Distribution
Notation:
Mean: x-bar
Standard Deviation:
s (lower case s)
The 68-95-99.7 Rule
The Empirical Rule
 In the normal
distribution with mean
 and standard
deviation :
 68% of the
observations fall within
1 of .
 95% of the
observations fall within
2 of .
 99.7% of the
observations fall within
3 of .
The standard normal distribution
All normal distributions are the same if we
measure in units of size  about the mean
 as center. Changing the units is called
standardizing.
X 
Z

z-score – tells how many standard
deviations the original observation falls
from the
 mean, and in which direction.
Standardizing and Z-scores
 Standardizing normal distributions make them all
the same. They are still normal.
 It produces a new variable that has the standard
normal distribution of N(0,1).
 When a score is expressed in standard deviation
units, it is referred to as a Z-score
EX: A score that is one standard deviation above the
mean has a Z-score of 1.
A score that is one standard deviation below the
mean has a Z-score of -1.
A score that is at the mean would have a Z-score of 0.
Why do we standardize?
This makes it possible to compare to
distributions easily.
Lets look at # 19 to better understand!
Remember:
X 
Z


Normal Distribution
Calculations
Since all normal distributions are normal
when we standardize, we can find the
areas under any normal curve from a
single table.
Table A (inside the front cover of text)
gives areas under the curve for standard
normal distribution.
Lets do #21 on p. 103 to better understand
how to use the table!
Assignment
Read through the end of the chapter.
Do #20, 22-26, 29